So, if I'm not wrong, the definition of the exterior derivative of a differential $k$-form is the differential $k\text{+}1$-form
$$\text{d}\omega = \omega_{i_1...i_k,i_{k+1}} \text{d}x^{i_{k\text{+}1}}\wedge\text{d}x^{i_1}\wedge\cdots\wedge\text{d}x^{i_k}.$$
When I was introduced to the concept of the covariant derivative, I was told that the partial derivative (the one used to define the exterior derivative of differential forms) doesn't produce objects whose components transform as tensor components would under a change of basis, so... how does that not contradict the idea of the exterior derivative mapping differential $k$-forms to differential $k\text{+}1$-forms? Am I missing something or am I using wrong definitions?